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April  2011, 30(1): 137-167. doi: 10.3934/dcds.2011.30.137

## Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations

 1 Franklin W. Olin College of Engineering, 1000 Olin Way, Needham, MA 02492-1200, United States 2 Gettysburg College, Department of Mathematics, 300 North Washington St., Gettysburg, PA 17325-1400, United States

Received  October 2009 Revised  November 2010 Published  February 2011

We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some cellular neural networks, monotone systems, and semi-discretizations for hyperbolic conservation laws with a source term. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.
Citation: Aaron Hoffman, Benjamin Kennedy. Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 137-167. doi: 10.3934/dcds.2011.30.137
##### References:
 [1] P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice,, SIAM J. Math. Anal., 35 (2003), 520. doi: 10.1137/S0036141000374002. [2] S. Benzoni-Gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws,, Phys. D, 115 (1998), 109. doi: 10.1016/S0167-2789(97)00225-X. [3] S. Bianchini, BV solutions of semidiscrete upwind scheme,, in, (2003), 135. [4] J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,, SIAM J. Appl. Math, 59 (1999), 455. [5] J. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits and Systems, 35 (1998), 1257. doi: 10.1109/31.7600. [6] J. O. Chua and L. Yang, Cellular neural networks: applications,, IEEE Trans. Circuits and Systems, 35 (1998), 1273. doi: 10.1109/31.7601. [7] J. L. Daleckii and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Space,", Translated from the Russian by S.Smith, 43 (). [8] O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, "Delay Equations: Functional- Complex- and Nonlinear Analysis,", volume \textbf{110} of Applied Mathematical Sciences, 110 (1995). [9] U. Ebert, W. Van Saarloos and L. A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations,, European J. Appl. Math, 13 (2002), 53. doi: 10.1017/S0956792501004673. [10] R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335. [11] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", volume \textbf{99} of Applied Mathematical Sciences, 99 (1993). [12] D. Henry, Small solutions of linear autonomous functional differential equations,, J. Differential Equations, 8 (1970), 494. doi: 10.1016/0022-0396(70)90021-5. [13] A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning,, J. Dynam. Differential Equations, 22 (2010), 79. doi: 10.1007/s10884-010-9157-2. [14] S.-S. Hsu and C.-H. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system,, J. Differential Equations, 164 (2000), 431. doi: 10.1006/jdeq.2000.3770. [15] S.-S. Hsu, C.-H. Lin and W. Shen, Traveling waves in cellular neural networks,, International Journal of Bifurcation and Chaos, 9 (1999), 1307. doi: 10.1142/S0218127499000912. [16] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776. doi: 10.1137/070703016. [17] J. Keener, Propagation and its failure in coupled systems of discrete excitable cells,, SIAM J. Appl. Math., 47 (1987), 556. doi: 10.1137/0147038. [18] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et som application a un probleme biologique,, Bull. Universite d'Etat a Moscou Ser. Int., 1 (1937), 1. [19] T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83. doi: 10.1007/s10998-008-5083-x. [20] B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323. doi: 10.1007/s00285-008-0175-1. [21] J. Mallet-Paret, Morse decompositions for delay-differential equations,, J. Differential Equations, 72 (1988), 270. doi: 10.1016/0022-0396(88)90157-X. [22] J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. doi: 10.1023/A:1021889401235. [23] J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type,, in, 1822 (2003), 231. [24] J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay,, J. Differential Equations, 125 (1996), 441. doi: 10.1006/jdeq.1996.0037. [25] C. Mascia, Qualitative behavior of conservation laws with reaction term and nonconvex flux,, Quart. Appl. Math., 58 (2000), 739. [26] R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations,, Ann. Mat. Pura Appl., 4 (1974), 263. [27] L. A. Peletier and J. A. Rodriguez, Fronts on a lattice,, Differential Integral Equations, 17 (2004), 1013. [28] W. Rudin, "Principles of Mathematical Analysis,", McGraw-Hill, (1976). [29] D. Serre, Discrete shock profiles: Existence and stability,, in, 1911 (2007), 79. [30] W. van Saarloos, Front Propagation into unstable states,, Phys. Rep., 386 (2003), 29. doi: 10.1016/j.physrep.2003.08.001. [31] R. van Zon, H. van Beijeren and Ch. Dellago, Largest Lyapunov exponent for many particle systems at low densities,, Phys. Rev. Lett., 80 (1998), 2035. doi: 10.1103/PhysRevLett.80.2035. [32] H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. doi: 10.1137/0513028.

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##### References:
 [1] P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice,, SIAM J. Math. Anal., 35 (2003), 520. doi: 10.1137/S0036141000374002. [2] S. Benzoni-Gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws,, Phys. D, 115 (1998), 109. doi: 10.1016/S0167-2789(97)00225-X. [3] S. Bianchini, BV solutions of semidiscrete upwind scheme,, in, (2003), 135. [4] J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,, SIAM J. Appl. Math, 59 (1999), 455. [5] J. O. Chua and L. Yang, Cellular neural networks: Theory,, IEEE Trans. Circuits and Systems, 35 (1998), 1257. doi: 10.1109/31.7600. [6] J. O. Chua and L. Yang, Cellular neural networks: applications,, IEEE Trans. Circuits and Systems, 35 (1998), 1273. doi: 10.1109/31.7601. [7] J. L. Daleckii and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Space,", Translated from the Russian by S.Smith, 43 (). [8] O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, "Delay Equations: Functional- Complex- and Nonlinear Analysis,", volume \textbf{110} of Applied Mathematical Sciences, 110 (1995). [9] U. Ebert, W. Van Saarloos and L. A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations,, European J. Appl. Math, 13 (2002), 53. doi: 10.1017/S0956792501004673. [10] R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 335. [11] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations,", volume \textbf{99} of Applied Mathematical Sciences, 99 (1993). [12] D. Henry, Small solutions of linear autonomous functional differential equations,, J. Differential Equations, 8 (1970), 494. doi: 10.1016/0022-0396(70)90021-5. [13] A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning,, J. Dynam. Differential Equations, 22 (2010), 79. doi: 10.1007/s10884-010-9157-2. [14] S.-S. Hsu and C.-H. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system,, J. Differential Equations, 164 (2000), 431. doi: 10.1006/jdeq.2000.3770. [15] S.-S. Hsu, C.-H. Lin and W. Shen, Traveling waves in cellular neural networks,, International Journal of Bifurcation and Chaos, 9 (1999), 1307. doi: 10.1142/S0218127499000912. [16] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations,, SIAM J. Math. Anal., 40 (2008), 776. doi: 10.1137/070703016. [17] J. Keener, Propagation and its failure in coupled systems of discrete excitable cells,, SIAM J. Appl. Math., 47 (1987), 556. doi: 10.1137/0147038. [18] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et som application a un probleme biologique,, Bull. Universite d'Etat a Moscou Ser. Int., 1 (1937), 1. [19] T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83. doi: 10.1007/s10998-008-5083-x. [20] B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323. doi: 10.1007/s00285-008-0175-1. [21] J. Mallet-Paret, Morse decompositions for delay-differential equations,, J. Differential Equations, 72 (1988), 270. doi: 10.1016/0022-0396(88)90157-X. [22] J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type,, J. Dynam. Differential Equations, 11 (1999), 1. doi: 10.1023/A:1021889401235. [23] J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type,, in, 1822 (2003), 231. [24] J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay,, J. Differential Equations, 125 (1996), 441. doi: 10.1006/jdeq.1996.0037. [25] C. Mascia, Qualitative behavior of conservation laws with reaction term and nonconvex flux,, Quart. Appl. Math., 58 (2000), 739. [26] R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations,, Ann. Mat. Pura Appl., 4 (1974), 263. [27] L. A. Peletier and J. A. Rodriguez, Fronts on a lattice,, Differential Integral Equations, 17 (2004), 1013. [28] W. Rudin, "Principles of Mathematical Analysis,", McGraw-Hill, (1976). [29] D. Serre, Discrete shock profiles: Existence and stability,, in, 1911 (2007), 79. [30] W. van Saarloos, Front Propagation into unstable states,, Phys. Rep., 386 (2003), 29. doi: 10.1016/j.physrep.2003.08.001. [31] R. van Zon, H. van Beijeren and Ch. Dellago, Largest Lyapunov exponent for many particle systems at low densities,, Phys. Rev. Lett., 80 (1998), 2035. doi: 10.1103/PhysRevLett.80.2035. [32] H. F. Weinberger, Long-time behavior of a class of biological models,, SIAM J. Math. Anal., 13 (1982), 353. doi: 10.1137/0513028.
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